the geography of compact irreducible holomorphic symplectic (or hyperkähler) varieties

Polarisation type

Associated to a Lagrangian fibration $X\to\mathbb{P}^n$ one can associate the polarisation type, which describes the polarisation induced by the restriction of an appropriate Kähler class on $X$ to the abelian varieties which appear as generic fiber, see [Theorem 1.1, MR3556822].

For OG6- resp. OG10-type, because the monodromy group is as large as possible, there is only a single type possible. For K3[n]-type there is still only a single type, despite the monodromy not being as large as possible, whilst for Kumn-type multiple types can arise.

See [Theorem 1.1, MR3556822].
Multiple types can appear, see [Theorem 1.1, MR3848435].
See [Corollary 1.3, MR4197280].
See [Theorem 2.2, 2010.12511v2].
dimension K3 K3[n]-type Kumn-type OG6 OG10
2 $(1)$
4 $(1)$ $(1,3)$
6 $(1)$ $(1,1,4),(1,2,2)$ $(1,2,2)$
8 $(1)$ $(1,1,1,5)$
10 $(1)$ $(1,1,1,1,6)$ $(1,1,1,1,1)$
12 $(1)$ $(1,1,1,1,1,7)$
14 $(1)$ $(1,1,1,1,1,1,8),(1,1,1,1,1,2,4)$
16 $(1)$ $(1,1,1,1,1,1,1,9),(1,1,1,1,1,1,3,3)$
18 $(1)$ $(1,1,1,1,1,1,1,1,10)$
20 $(1)$ $(1,1,1,1,1,1,1,1,1,11)$

Wieneck, Benjamin. "On polarization types of Lagrangian fibrations." In: Manuscripta Math. 151 (2016), pp. 305–327. doi:10.1007/s00229-016-0845-z
Wieneck, Benjamin. "Monodromy invariants and polarization types of generalized Kummer fibrations." In: Math. Z. 290 (2018), pp. 347–378. doi:10.1007/s00209-017-2020-y
Mongardi, Giovanni and Rapagnetta, Antonio. "Monodromy and birational geometry of O'Grady's sixfolds." In: J. Math. Pures Appl. (9) 146 (2021), pp. 31–68. doi:10.1016/j.matpur.2020.12.006
Mongardi, Giovanni and Onorati, Claudio. "Birational geometry of irreducible holomorphic symplectic tenfolds of O'Grady type". arXiv:arXiv:2010.12511